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Robert C. Kirby A general approach to transforming finite elements SMAIJournal of computational mathematics, 4 (2018), p. 197224, doi: 10.5802/smaijcm.33 Article PDF Class. Math.: 65N30 Keywords: Finite element method, basis function, pullback Abstract The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However, many famous finite elements such as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence needed to work with a reference element in the standard way. This paper gives a generalized approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under pushforward. This approach, developed through a sequence of examples of increasing complexity, requires one to study the relationship between the function space and degrees of freedom, or nodes, on a generic cell and the transformation of the corresponding entities on a reference cell. When the function space is preserved under mapping, one must be able to express the pushedforward finite element nodes as linear combinations of the reference element finite element nodes. The transpose of this linear transformation maps the pullback of the reference element basis functions to the desired finite element basis functions. Generically, developing this transformation for elements such as Morley and Hermite involves completing the set of finite element nodes, although the process is simplified in concrete ways when the finite elements form affine or affineinterpolation equivalent families such as Lagrange or Hermite. 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